10 Standard Lesson Notes Tuesday 1st September
Quick review of set builder notation and plotting intervals on number lines.
Another way of identifying an interval:
−3 ≤ 𝑥 < 4 can also be written as 𝑥 ∈ [−3,5[
Similarly 5 < 𝑥 < 8 can be written as 𝑥 ∈ ]5,8[
Venn Diagrams
A Venn Diagram has a rectangle to represent the Universal set, and circles to
represent sets within it.
e.g. Let ℰ = {𝑥|𝑥 ≤ 12, 𝑥 ∈ ℤ} and 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3}
This can be shown using a Venn diagram:
ℰ
1 2 4 5
3
6
19
12
7 8 10
11
A
This makes it easy to see which elemets are members of A, and which are
members of A’.
𝐴 = {3, 6, 9, 12} and 𝐴′ = {1, 2, 4, 5, 7, 8 10,11}
It is easy to count the elements too:
n(A)=4 and n(A’) = 8
There may be more sets.
e.g. Let ℰ = {𝑥|𝑥 ≤ 20, 𝑥 ∈ ℤ}, 𝐴 = {𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟𝑠} and 𝐵 = {𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 4}
ℰ
𝐴
1 3 5
2 6 10 𝐵
7 9 11 14
13 15 17
19
18
4 8 12
16 20
Now it is easy to see that 𝐵 ⊆ 𝐴.
Ex2D page 67
Intersection and Union
The union of two sets is written 𝐴⋃𝐵.
It includes all the elements that are in A or B or in both.
The intersection of two sets is written 𝐴⋂𝐵.
It includes all the elements that are in both set A and set B.
e.g. e.g. Let ℰ = {𝑥|𝑥 ≤ 12, 𝑥 ∈ ℤ} , 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3}
and 𝐵 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 2}
𝐴⋃𝐵 = {2, 3, 4, 6, 8, 9, 10, 12}
and 𝐴 ∩ 𝐵 = {6, 12}
Disjoint sets
Two sets are said to be disjoint sets if they have no elements in common.
i.e. 𝐴 ∩ 𝐵 = 𝜙
Page 69 Ex 2E.1